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A166293 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at even level (0<=k<=n-1; U=(1,1), D=(1,-1)).

%I A166293 #2 Mar 30 2012 17:36:23
%S A166293 1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,13,12,5,1,1,6,22,28,18,6,1,1,7,35,
%T A166293 59,50,25,7,1,1,8,54,114,124,80,33,8,1,1,9,82,210,279,226,119,42,9,1,
%U A166293 1,10,124,374,592,576,375,168,52,10,1,1,11,188,653,1199,1374,1062,582,228,63
%N A166293 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at even level (0<=k<=n-1; U=(1,1), D=(1,-1)).
%C A166293 Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
%C A166293 T(n,0)=1.
%C A166293 Sum(k*T(n,k), k=0..n-1)=A166294(n).
%F A166293 The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).
%e A166293 T(4,2)=3 because we have UDU(UD)(UD)D, U(UD)(UD)DUD, and U(UD)DU(UD)D (the even-level peaks are shown between parentheses).
%e A166293 Triangle starts:
%e A166293 1;
%e A166293 1,1;
%e A166293 1,2,1;
%e A166293 1,3,3,1;
%e A166293 1,4,7,4,1;
%e A166293 1,5,13,12,5,1.
%p A166293 p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(t = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], s, j), j = 0 .. n-1) end do; # yields sequence in triangular form
%Y A166293 Cf. A004148, A166291, A166292, A166294
%K A166293 nonn,tabl
%O A166293 1,5
%A A166293 _Emeric Deutsch_, Oct 12 2009